psm12se wrote:How many three-digit odd integers less than 500 are there so that all the digits are not distinct?

A. 56

B. 88

C. 144

D. 200

E. 250

Integers with all 3 digits NOT distinct = total number of integers - integers with all 3 digits distinct.

Total number of integers:
To count consecutive integers:
Number of integers = biggest - smallest + 1.
Thus, from 100 to 499, inclusive:
Number of integers = 499 - 100 + 1 = 400.

Integers with all 3 digits distinct:
Number of options for the hundreds digit = 4. (1, 2, 3 or 4).
Number of options for the tens digit = 9. (Any digit 0-9 but the digit already used.)
Number of options for the units digit = 8. (Any digit 0-9 but the 2 digits already used.)
To combine these options, we multiply:
4*9*8 = 288.

Thus, from 100 to 499, inclusive:
Number of integers with all 3 digits NOT distinct = 400-288 = 112.

Of these 112 integers, exactly HALF will be odd:
112/2 = 56.

The correct answer is A.